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Transverse instability of a bunched beam in a ring accelerator is considered. An emphasis is given to an investigation of rigid modes at space charge dominating beam coupling impedance and about linear synchrotron oscillations. The theory is applied to the Fermilab Recycler [1] which is treated as a proton accumulator in context of Proton Plan II [2]. It is shown that the instability growth time can reach about 0.3 ms, and chromaticity about -7 to -10 is required to suppress it. The suppression by means of 26 MHz damper is considered also, and required parameters of the damper are provided. Arguments are adduced that obtained stability conditions are not only necessary but also sufficient, because other modes except the rigid ones are more stable

Recycler transverse instability in context of proton plan II